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A Frequency Analysis of Light TransportA Frequency Analysis of Light Transport
F. Durand, MIT CSAIL
N. Holzschuch, C. Soler, ARTIS/GRAVIR-IMAG INRIA
E. Chan, MIT CSAIL
F. Sillion, ARTIS/GRAVIR-IMAG INRIA
Illumination effectsIllumination effects
• Shadow boundaries:
© U. Assarsson 2005.
Point light source Area light source
Illumination effectsIllumination effects
• Indirect lighting is usually blurry:
Indirect lighting onlyDirect lighting only
Frequency aspects of light transportFrequency aspects of light transport
• Blurriness = frequency content
– Sharp variations: high frequency
– Smooth variations: low frequency
• All effects are expressed as frequency content:
– Diffuse shading: low frequency
– Shadows: introduce high frequencies
– Indirect lighting: tends to be low frequency
Problem statementProblem statement
• How does light interaction in a scene explain the frequency content?
• Theoretical framework:
– Understand the frequency spectrum of the radiance function
– From equations of light transport
Unified framework:Unified framework:
• Spatial frequency (e.g. shadows, textures)
• Angular frequency (e.g. blurry highlight)
Disclaimer: not Fourier opticsDisclaimer: not Fourier optics
• We do not consider wave optics, interference, diffraction
• Only geometrical optics
Fro
m [
Hec
ht]
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
Previous workPrevious work
• Vast body of literature:– Light field sampling
– Perceptually-based rendering
– Wavelets for Computer Graphics
– Irradiance caching
– Photon mapping
– …
• We focus on frequency analysis in graphics:– Light field sampling
– Reflection as a convolution
Light field samplingLight field sampling
[Chai et al. 00, Isaksen et al. 00, Stewart et al. 03]
– Light field spectrum as a function of object distance
– No BRDF, occlusion ignored
From [Chai et al. 2000]
Signal processing for reflectionSignal processing for reflection
[Ramamoorthi & Hanrahan 01, Basri & Jacobs 03]
• Reflection on a curved surface is a convolution
• Direction only
From [Ramamoorthi and Hanrahan 2001]
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
Our approachOur approach
• Light sources are input signal
• Interactions are filters/transforms
– Transport
– Visibility
– BRDF
– Etc.
Our approachOur approach
• Light sources are input signal
• Interactions are filters/transforms
– Transport
– Visibility
– BRDF
– Etc.
Light source signal
Our approachOur approach
• Light sources are input signal
• Interactions are filters/transforms
– Transport
– Visibility
– BRDF
– Etc.
Light source signal
Transport
Our approachOur approach
• Light sources are input signal
• Interactions are filters/transforms
– Transport
– Visibility
– BRDF
– Etc.
Light source signal
Signal 1
Transport
Our approachOur approach
• Light sources are input signal
• Interactions are filters/transforms
– Transport
– Visibility
– BRDF
– Etc.
Light source signal
Transport
Occlusion
Signal 2
Our approachOur approach
• Light sources are input signal
• Interactions are filters/transforms
– Transport
– Visibility
– BRDF
– Etc.
Light source signal
Transport
Occlusion
Signal 3
Transport
Our approachOur approach
• Light sources are input signal
• Interactions are filters/transforms
– Transport
– Visibility
– BRDF
– Etc.
Light source signal
Transport
Occlusion
Signal 4
Transport
BRDF
Local light fieldLocal light field
• 4D light field, around a central ray
• We study its spectrum during transport
Local light fieldLocal light field
• 4D light field, around a central ray
• We study its spectrum during transport
Local light fieldLocal light field
• 4D light field, around a central ray
• We study its spectrum during transport
Local light fieldLocal light field
• We give explanations in 2D
– Local light field is therefore 2D
• See paper for extension to 3D
Local light field parameterizationLocal light field parameterization
• Space and angle
space
angle
Central ray
Local light field representationLocal light field representation
• Density plot:
Area light source
Space
Ang
le
Local light fieldFourier spectrumLocal light fieldFourier spectrum
• We are interested in the Fourier spectrum of the local light field
• Also represented as a density plot
Local light field Fourier spectrumLocal light field Fourier spectrum
Spatial frequency
Ang
ular
freq
uenc
ySpectrum of an area light source:
Fourier analysis 101Fourier analysis 101
• Spectrum corresponds to blurriness:
– Sharpest feature has size 1/Fmax
• Convolution theorem:– Multiplication convolution
• Classical spectra: – Box sinc– Dirac constant
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Transport
• Occlusion
• BRDF
• Curvature
• Case studies
• Conclusions and future directions
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Transport
• Occlusion
• BRDF
• Curvature
• Case studies
• Conclusions and future directions
Transport in free spaceTransport in free space
Shear
Space Space
Ang
le
Ang
leShear
Spatial frequency
Ang
ular
freq
.
Spatial frequency
Ang
ular
freq
.
Transport ShearTransport Shear
• This is consistent with light field spectra[Chai et al. 00, Isaksen et al. 00]
From [Chai et al. 2000]
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Transport
• Occlusion
• BRDF
• Curvature
• Case studies
• Conclusions and future directions
Occlusion: multiplicationOcclusion: multiplication
• Occlusion is a multiplication in ray space
– Convolution in Fourier space
• Creates new spatial frequency content
– Related to the spectrum of the blockers
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Transport
• Occlusion
• BRDF
• Curvature
• Case studies
• Conclusions and future directions
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Transport
• Occlusion
• BRDF
• Curvature
• Case studies
• Conclusions and future directions
BRDF integrationBRDF integration
• Outgoing light:
– Integration of incoming light times BRDF
Outgoing lightIncoming light
BRDF
BRDF integrationBRDF integration
• Ray-space: convolution
– Outgoing light: convolution of incoming light and BRDF
– For rotationally-invariant BRDFs
• Fourier domain: multiplication
– Outgoing spectrum: multiplication of incoming spectrum and BRDF spectrum
BRDF in Fourier: multiplicationBRDF in Fourier: multiplication
=
• BRDF is bandwidth-limiting in angle
Example: diffuse BRDFExample: diffuse BRDF
• Convolve by constant:
– multiply by Dirac
– Only spatial frequencies remain
=
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Transport
• Occlusion
• BRDF
• Curvature
• Case studies
• Conclusions and future directions
Curved receiverCurved receiver
• Reduce to the case of a planar surface:
– “Unroll” the curved receiver
• Equivalent to changing angular content:
– Linear effect on angular dimension
– No effect on spatial dimension
• Shear in the angular dimension
Transforms: summaryTransforms: summary
Radiance/Fourier Effect
Transport Shear
Occlusion Multiplication/Convolution Adds spatial frequencies
BRDF Convolution/Multiplication Removes angular frequencies
Curvature Shear
More effects in paperMore effects in paper
• Cosine term (multiplication/convolution)
• Fresnel term (multiplication/convolution)
• Texture mapping (multiplication/convolution)
• Central incidence angle (scaling)
• Separable BRDF
• Spatially varying BRDF (semi-convolution)
• …and extension to 3D
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
Diffuse soft shadowsDiffuse soft shadows
• Decreasing blockers size:
– First high-frequencies increase
– Then only low frequency
– Non-monotonic behavior
Diffuse soft shadows (2) Diffuse soft shadows (2)
• Occlusion : convolution in Fourier
– creates high frequencies
– Blockers scaled down spectrum scaled up
Fourier space
v
(an
gle
)
x (space)
Fourier space
v
(an
gle
)x (space)
blocker spectrum
Diffuse soft shadows (3)Diffuse soft shadows (3)
• Transport after occlusion:
– Spatial frequencies moved to angular dimension
• Diffuse reflector:
– Angular frequencies are cancelled
Diffuse soft shadows (3)Diffuse soft shadows (3)
• Transport after occlusion:
– Spatial frequencies moved to angular dimension
• Diffuse reflector:
– Angular frequencies are cancelled
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
Adaptive shading samplingAdaptive shading sampling
• Monte-Carlo ray tracing
• Blurry regions need fewer shading samples
Adaptive shading samplingAdaptive shading sampling
• Per-pixel prediction of max. frequency (bandwidth)
– Based on curvature, BRDF, distance to occluder, etc.
– No spectrum computed, just estimate max frequency
Per-pixel bandwidth criterion
Adaptive shading samplingAdaptive shading sampling
• Per-pixel prediction of max. frequency (bandwidth)
– Based on curvature, BRDF, distance to occluder, etc.
– No spectrum computed, just estimate max frequency
Shading samples
OverviewOverview
• Previous work
• Our approach:
– Local light field
– Transformations on local light field
• Case studies:
– Diffuse soft shadows
– Adaptive shading sampling
• Conclusions and future directions
ConclusionsConclusions
• Unified framework:
– For frequency analysis of radiance
– In both space and angle
– Simple mathematical operators
– Extends previous analyses
• Explains interesting lighting effects:
– Soft shadows, caustics
• Proof-of-concept:
– Adaptive sampling
Future workFuture work
• More experimental validation on synthetic scenes
• Extend the theory:
– Bump mapping, microfacet BRDFs, sub-surface scattering…
– Participating media
• Applications to rendering:
– Photon mapping
– Spatial sampling for PRT
– Revisit traditional techniques
• Applications to vision and shape from shading
AcknowledgmentsAcknowledgments
• Jaakko Lehtinen
• Reviewers of the MIT and ARTIS graphics groups
• Siggraph reviewers
• This work was supported in part by:– NSF CAREER award 0447561 Transient Signal Processing for Realistic
Imagery
– NSF CISE Research Infrastructure Award (EIA9802220)
– ASEE National Defense Science and Engineering Graduate fellowship
– INRIA Équipe associée
– Realreflect EU IST project
– MIT-France
Other bases?Other bases?
• We’re not using Fourier as a function basis
– Don’t recommend it, actually
– Just used for analysis, understanding, predictions
• Results are useable with any other base:
– Wavelets, Spherical Harmonics, point sampling, etc
– Max. frequency translates in sampling rate
• Analysis relies on Fourier properties:
– Especially the convolution theorem
Why Local Light Field?Why Local Light Field?
• Linearization:
– tan
– Curvature
• Local information is what we need:
– Local frequency content, for local sampling
– Local properties of the scene (occluders, curv.)
Reflection on a surface: Full summaryReflection on a surface: Full summary
• Angle of incidence
• Curvature
• Cosine/Fresnel term
• Mirror re-parameterization
• BRDF
• Curvature
Reflection on a surface: Full summaryReflection on a surface: Full summary
• Angle of incidence: scaling
• Curvature: shear in angle
• Cosine/Fresnel term: multiplication/convolution
• Mirror re-parameterization
• BRDF: convolution/multiplication
• Curvature: shear in angle